This is not answerable with satisfactory comprehensiveness. The time before settlement and adoption of Christianity around 1000 is just too poorly documented.
But apart from what Wikipedia presents in Old Hungarian script and some information in History of the Hungarian language, this is probably the secondary source coming most closely to what the question seems to inquire about the earliest traces in condensed form:
The history of the Magyars before they settled in the territory that was to be their country has remained in obscurity, which no research work could clear up so far. Only some moments of the early history of the Magyars have been elucidated by conclusions drawn from diverse sources, linguistic, ethnographic, or archeologic facts. Thus, it can be taken for certain that the structure and basic vocabulary of the Hungarian language is of Finno-Ugrian origin, and the nearest relatives of the Magyars, as far as their language is concerned, are two peoples now living in Siberia: the Manshi (Voguls) and the Chanti (Ostyaks). It follows from this that one part of the later Himgarian people must have lived in close contact with these two peoples some where near the Urals. But it is equally clear that Hungarian had also absorbed Turkic linguistic elements before the final settlement. Later, at the time of the Conquest, when the Magyars had already emerged as a distinct nation, they had several Turkish characteristics; and their oldest names denoting tribes, persons, or dignitaries are partly of Turkish origin. All this proves that the Finno-Ugrian elements of the Himgarian nation must have mingled with Turkish elements even before their final settlement. The hypothesis is also supported by the fact that sometimes the Magyars are referred to as Turks in the earliest sources.
The Byzantine source that makes the first mention of the Magyars calls them “Ungroi” (Ουγγροι), “Turkoi” (Τουρχοι), or “Unnoi” (Ουννοι). The first of these is surely identical with the name that had been used by nations in contact with the Magyars, and that is used even today in most European languages. Its variants are: the Church Slavic “Ugry” (Оугры),the Russian “Vengry” (Венгры), the Greek “Ungroi” or “Ungaroi” (Ουγγροι, Ουγγαροι), the Latin “Ungari” or “Hungari”, the German “Ungar”, the French “Hongrois”, the Italian “Ungheresi”, the English “Hungarian”, etc.
Linguists have pointed out that all these names originate from the ethnic name “Onogur”, meaning “ten Ugors”, a name of Turkish origin, later adopted by the Slavonic languages. The name “Onogur” denoting “Hungarian" occurs in documents dated from the 5th to 8th centuries A.D. It has not been adequately explained how this transfer was made. Some consider the two peoples to be completely identical; others assume a close contact between them; still others maintain that the transfer was made by Slavs at the time when the two nations occupied the same territory. Whatever the origin of this transfer, all students of early Hungarian history must take the Onogurs into consideration. All facts prove that different ethnic groups went into the formation of the Hungarian people; an examination of these groups helps to localize the earliest homes of the Magyars, still in the process of formation.
According to Byzantine sources the Ogur tribes moved from the North to the northern coast of the Black Sea after the fall of the Hun empire. A work of the rhetor Priscus contains a description of the population movement, during which the Savirs wore driven by the Avars from their territory. The Saragurs, the Ogurs, and the Onogurs, having been expelled by the Savirs, sent representatives to Byzantium about 463. In the case of the Saragurs our source clearly defines the aim of the mission: as many other nations, they wished to obtain the favour of the Byzantine court, in order to secure the territory occupied by them in the neighbourhood of the Byzantine Empire. The delegates were welcomed and as usual given presents at the court. From innumerable similar cases one can draw the conclusion that by having done so the Byzantine court complied with their requests and granted them the annual presents on condition that they join the network of peoples in alliance with Byzantium. These peoples were expected to defend the imperial frontiers and, in case of necessity, fight the enemies of the Empire. Although no clear indication can be found in documents, we must suppose that the Onogurs played such a role in the vicinity of Lake Maeotis (later known as the Sea of Azov), the territory where later sources put their home. This is supported by a reference to an occasion (the date is not defined in the sources) when the Onogurs, probably encouraged by Byzantium, engaged in battle with the people of Colchis, that is the Lazi, who occupied the eastern coast of the Black Sea, to the south of the Caucasus, and were under the political influence of Pereia up to the 6th century.
Gyula Moravcsik: "Byzantium and the Magyars", Hakkert: Amsterdam, 1970.
An nice overview of another reason, apart from sparse documentary evidence – namely fierce want for nationalist, and thus fictional, interpretation of the past of a people – for otherwise 'academic' historical research is given is in Charles R. Bowlus: "Franks, Moravians, and Magyars: The Struggle for the Middle Danube 788-907", Middle Ages Series, University of Pennsylvania Press, 1995.
Even when looking at the Byzantine sources, which seem to be overlooked a bit in Western scholarship, this is still a somewhat Western perspective. This is complemented with the Arabic sources, with an early example of ethnography of Eastern Europe:
The name al-Jayhānī refers to a geographer at the court of the Samanids in the 10th century. Several members of the al-Jayhānī family acquired high positions and had an education that allowed them to pursue serious literary activities.
The first family member who gained great reputation and rose to a higher state position was Abū Muḥammad ibn Aḥmad ʿAbdallāh al-Jayhānī. In 913, he was appointed guardian of the 8-year-old Samanid emir Naṣr ibn Aḥmad (914–943), and was wazīr during the emir’s reign.
The form majghir or majghar مجغر has usually been reconstructed from the authors’ manuscripts of the Jayhānī tradition. The ethnonym is found once ا فریة al-mujf.riyya, six times as المُجفریة as the title of the chapter in the form al-m.jf.riyya, and indeed once as المحفریة al-m.ḥf.riyya in the manuscripts of Ibn Rusta. Although the shape m.jgh/f.riyān can be found four times, it cannot be decided whether the reading should be gh غ or f ف, for the forms m.ḥf.riyān and m.jf.riyān occur three times and twice, respectively, in Gardīzī’s Cambridge manuscript, and finally occurs once as ىجغو ن x.jgh.ūyān with an uncertain initial and with ū و instead of r ر.
The Oxford manuscript of Gardīzī contains the uncertain reading m.jgh/f.riyān five times, m.ḥf.riyān four times, and the eth- nonym x.jf.riyān once. Al-Bakrī has the form al-m.ḥ/jf.riyya. Abūʾl-Fidāʾ gave the exact reading of the ethnonym: “with mīm and jīm and ghayn with a diacritical point and rāʾ without diacritical points and a letter with two points below ( y) then hāʾ at the end,” i.e. m.jgh.riyya. The uncertain form m.jgh/f.rī appears twice and m.jf.rī once in Ḥudūd al-ʿālam. Al-Marwazī recorded the form al- m.ḥf.riyya first, then al-m.ḥʿ.riyya three times without diacritical points, and finally al-m.jʿ.riyya. There are new variants in the late Persian Shukrallāh and its Turkic translations. The author of the Bahjat at-tawārīḥ recorded m.ḥr.q.h, but there is another variant in the Turkish translation of the form m.ḥt.rq.h.
The relationship among the different forms can be reconstructed in most cases. The final -iyya in the Arabic texts (Ibn Rusta, al-Bakrī and al-Marwazī) is a well-known composite suffix. Its first component, the -ī, forms an adjective from a noun, while the -a is the feminine ending as well as an abstract noun or collective suffix, which was used similarly in other ethnonyms. The ethnonyms in Persian end in -ī. The general rule is reflected in the Persian Ḥudūd al-ʿālam. Gardīzī completed this form with the Persian plural -ān: m.jf.riyān. According to Nyitrai both forms can be explained from the Arabic al-m.jf.riyya.
The letter m at the beginning of the word of the ethnonym is well-attested, the only exception being the form x.jf.riyān of Gardīzī, and the uncertainty of the reading is easily explained. The copyist writing the Arabic form al-m.jf.r in the Persian translation probably could not identify the letters lām mīm but wanted to indicate that the word began with a consonant (الىجفر~ ا فر al-m.jf.r ~ al-x.jf.r). It may therefore be regarded as an error of the copyist.
There are two consonants in the middle of the ethnonym. The first may be read as ḥ or j ج or ح, differing solely in a diacritical point. Identifying the second consonant, however, is more complicated. Ibn Rusta clearly used the letter f, and similarly Gardīzī has f five times, but another five times the readings are uncertain and either f or gh ف or غ can be reconstructed. Referring to the Ḥudūd al-ʿālam and the works of al-Marwazī and al-Bakrī a similar conclusion may be drawn: some readings as f are certain, but both readings are possible in other instances. The similarity of the two letter forms seems to have been a cause of uncertainty.
István Zimonyi: "Muslim Sources on the Magyars in the Second Half of the 9th Century. The Magyar Chapter of the Jayhānī Tradition", East Central and Eastern Europe in the Middle Ages, 450–1450, 35, Brill: Leiden, Boston, 2016. (DOI)
In a modernised summary:
The ethnonym ‘Magyar,’ for the first time attested in writing (done in Greek letters) in 810, was the name of one of the Magyar tribes or clans, the Megyers. Those who claim it as a Finno-Ugric word, propose that magy is an ethnic name of unclear etymology, and the Finno-Ugric particle –eri means ‘men,’ or more generally ‘people.’ Recently, it became accepted that ‘Magyar’ and Megyer is related to the self-ethnonym of Siberia’s Finno-Ugric ethnic group, the Mansis (Voguls). Following the breakup of the hypothetic Ugric linguistic community in 1000 BCE, the ancestors of the Megyers and the Mansis entered into a long- lasting contact with the Iranian-speaking population in the areas north of the Aral and Caspian Seas. In the case of the Megyers, the contact lasted until 600 CE. Hence, it is probable that both ethnonyms stem from an Iranian linguistic loan meaning ‘human.’ The term ‘Hungary’ began to appear in the 8th-century Frankish documents done in Latin. It is derived from the Turkic term onogur, in which on means ‘ten,’ and ogur ‘arrows.’ This was the name of the coalition of seven Finno-Ugric (Magyar) and three Turkic clans (tribes) that entered the Danubian basin in the late 9th century. With time, the smaller Turkic element was assimilated with the Finno-Ugric-speakers, hence from the early times, the Magyars’ self-ethnonym was Magyar. However, the name of the Onogur coali- tion recorded time and again in Latin documents (Ungari, Ungri, and Hungari), remained the name under which the Magyars and their state was known to other peoples in Europe. (Abondolo 1998: 387–390, 453; Benko ̋, 1970: 816, 1976: 1025; Középiskolai történelmi atlasz 1996: 18; Melnychuk et al. 1989: 357).
Tomasz Kamusella: "The Politics of Language and Nationalism in Modern Central Europe", Palgrave MacMillan: Basingstoke, New York, 2009.